Problem: Determine how many solutions exist for the system of equations. ${3x-y = -1}$ ${-4x+y = -5}$
Solution: Convert both equations to slope-intercept form: ${3x-y = -1}$ $3x{-3x} - y = -1{-3x}$ $-y = -1-3x$ $y = 1+3x$ ${y = 3x+1}$ ${-4x+y = -5}$ $-4x{+4x} + y = -5{+4x}$ $y = -5+4x$ ${y = 4x-5}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 3x+1}$ ${y = 4x-5}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.